Relativistic Mass Problem

In the spectator's frame, it seems as though both Einstein and Lorentz have relative velocities from 0.6c and -0.6c.

Then in Einstein's frame, Lorentz's ball has more mass.

Based on the above assumption, Lorentz's ball has 2.125 times more mass than Einstein's ball, since the relative velocity between Einstein and Lorentz is 0.88c.

However, the spacetime diagram only deals with things in a one-dimensional space. It seems as though they haven't taken into account the vertical velocity of Lorentz's ball. So is this an appropriate way to calculate the relativistic mass of Lorentz's ball?

There is also another catch. If we consider that 2-D scene playing out in 1-D space, Einstein and Lorentz's balls collide, reflect, Einstein collects Lorentz's ball and Lorentz collects Einstein's ball.

In Einstein's frame, his ball is at rest w.r.t. to him until Lorentz's ball knocks it away. There is a complete transfer of momentum from Lorentz's ball to Einstein's ball.

This implies that in either frame, both balls have the same mass, which is contrary to the interpretation in the video.

In the spectator's frame, it seems as though both Einstein and Lorentz have relative velocities from 0.6c and -0.6c.

Then in Einstein's frame, Lorentz's ball has more mass.

Based on the above assumption, Lorentz's ball has 2.125 times more mass than Einstein's ball, since the relative velocity between Einstein and Lorentz is 0.88c.

However, the spacetime diagram only deals with things in a one-dimensional space. It seems as though they haven't taken into account the vertical velocity of Lorentz's ball. So is this an appropriate way to calculate the relativistic mass of Lorentz's ball?

Yes, it is. The two balls have the same vertical speed so any increase in mass due to their vertical speeds will be the same.

There is also another catch. If we consider that 2-D scene playing out in 1-D space, Einstein and Lorentz's balls collide, reflect, Einstein collects Lorentz's ball and Lorentz collects Einstein's ball.

I don't understand what you mean by that. How could they catch each other's ball int 1, 2, or 3 D?

In Einstein's frame, his ball is at rest w.r.t. to him until Lorentz's ball knocks it away. There is a complete transfer of momentum from Lorentz's ball to Einstein's ball.

No, it E's ball is NOT at rest with respect to him.

This implies that in either frame, both balls have the same mass, which is contrary to the interpretation in the video.

Perhaps I am misunderstanding what you are saying here but I don't see how that follows.